Comparing Fractions

When two fractions have the same denominator, their pieces are the same size, so the one with more pieces is bigger: $\frac{7}{9} > \frac{5}{9}$. Compare the tops and you are done.

Different denominators need a common one first. Compare $\frac{2}{3}$ and $\frac{3}{5}$: rename both over 15, as you did at the start of this chapter. $\frac{2}{3}$ is $\frac{10}{15}$ and $\frac{3}{5}$ is $\frac{9}{15}$. Now the pieces match, and 10 fifteenths beat 9, so $\frac{2}{3} > \frac{3}{5}$.

A shortcut does the same job: cross products. Multiply each top by the other fraction's bottom: $2\cdot 5$ is 10 and $3\cdot 3$ is 9. These are exactly the new tops over 15, so the fraction whose top gave the bigger product is the bigger fraction.